Classical Algorithms from Quantum and Arthur-Merlin Communication Protocols Lijie Chen Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] Ruosong Wang Carnegie Mellon University, Pittsburgh, PA, USA [email protected] Abstract In recent years, the polynomial method from circuit complexity has been applied to several√ funda- mental problems and obtains the state-of-the-art running times (e.g., R. Williams’s n3/2Ω( log n) time algorithm for APSP). As observed in [Alman and Williams, STOC 2017], almost all appli- cations of the polynomial method in algorithm design ultimately rely on certain (probabilistic) low-rank decompositions of the computation matrices corresponding to key subroutines. They suggest that making use of low-rank decompositions directly could lead to more powerful algo- rithms, as the polynomial method is just one way to derive such a decomposition. Inspired by their observation, in this paper, we study another way of systematically construct- ing low-rank decompositions of matrices which could be used by algorithms – communication pro- tocols. Since their introduction, it is known that various types of communication protocols lead to certain low-rank decompositions (e.g., P protocols/rank, BQP protocols/approximate rank). These are usually interpreted as approaches for proving communication lower bounds, while in this work we explore the other direction. We have the following two generic algorithmic applications of communication protocols: Quantum Communication Protocols and Deterministic Approximate Counting. Our first connection is that a fast BQP communication protocol for a function f implies a fast deterministic additive approximate counting algorithm for a related pair counting problem. Applying known BQP communication protocols, we get fast deterministic additive approximate counting algorithms for Count-OV (#OV), Sparse Count-OV and Formula of SYM circuits. In particular, our approximate counting algorithm for #OV runs in near-linear time for all dimensions d = o(log2 n). Previously, even no truly-subquadratic time algorithm was known for d = ω(log n). Arthur-Merlin Communication Protocols and Faster Satisfying-Pair Algorithms. Our second connection is that a fast AMcc protocol for a function f implies a faster-than- bruteforce algorithm for f-Satisfying-Pair. Using the classical Goldwasser-Sisper AM protocols for approximating set size, we obtain a new algorithm for approximate Max-IPn,c log n in time n2−1/O(log c), matching the state-of-the-art algorithms in [Chen, CCC 2018]. We also apply our second connection to shed some light on long-standing open problems in communication complexity. We show that if the Longest Common Subsequence (LCS) problem admits a fast (computationally efficient) AMcc protocol (polylog(n) complexity), then polynomial- 1−δ size Formula-SAT admits a 2n−n time algorithm for any constant δ > 0, which is conjectured to be unlikely by a recent work [Abboud and Bringmann, ICALP 2018]. The same holds even for a fast (computationally efficient) PHcc protocol. 2012 ACM Subject Classification Theory of computation → Communication complexity Keywords and phrases Quantum communication protocols, Arthur-Merlin communication pro- tocols, approximate counting, approximate rank © Lijie Chen and Ruosong Wang; licensed under Creative Commons License CC-BY 10th Innovations in Theoretical Computer Science (ITCS 2019). Editor: Avrim Blum; Article No. 23; pp. 23:1–23:20 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 23:2 Classical Algorithms from Quantum and Arthur-Merlin Communication Protocols Digital Object Identifier 10.4230/LIPIcs.ITCS.2019.23 Related Version A full version of the paper is available at https://arxiv.org/abs/1811. 07515. Acknowledgements The first author is grateful to Josh Alman, Chi-Ning Chou, Mika Göös, and Ryan Williams for helpful discussions during this work. We are grateful to anonymous reviewers for many helpful and inspiring comments on this paper. In particular, we thank one anonymous reviewer for pointing out that Theorem 5 can be improved using the polynomial method. 1 Introduction Recent works have shown that the polynomial method, a classical technique for proving circuit lower bounds [41, 45], can be useful in designing efficient algorithms [48, 50, 6, 10, 8, 36, 7]. At a very high level, these algorithms proceed as follows: (1) identify a key subroutine of the core algorithm which has a certain low-degree polynomial representation; (2) replace that subroutine by the corresponding polynomials, and reduce the whole problem to a certain batched evaluation problem of sparse polynomials; (3) embed that polynomial evaluation prob- lem to multiplication of two low-rank (rectangular) matrices, and apply the fast rectangular matrix multiplication algorithm [26]. As [9] point out. In term of step (3), these algorithms are ultimately making use of the fact that the corresponding matrices of some circuits or subroutines have low probabilistic rank.[9] suggest that the probabilistic rank, or various low-rank decompositions of matrices in general1, could be more powerful than the polynomial method, and lead to more efficient algorithms, as the polynomial method is just one way to construct them. It has been noted for a long time that communication protocols are closely related to various notions of rank of matrices. To list a few: deterministic communication complexity is lower bounded by the logarithm of the rank of the matrix [37]; quantum communication complexity is lower bounded by the logarithm of the approximate rank of the matrix [16, 19]; UPP communication complexity is equivalent to the logarithm of the sign-rank of the matrix [40]. These connections are introduced (and usually interpreted) as methods for proving communication complexity lower bounds (see, e.g. the survey by Lee and Shraibman [35]), but they can also be interpreted in the other direction, as a way to systematically construct low-rank decompositions of matrices. In this paper, we explore the connection between different types of communication protocols and low-rank decompositions of matrices and establish several applications in algorithm design. For all these connections, we start with an efficient communication protocol for a problem F , which implies an efficiently constructible low-rank decomposition of the corresponding communication matrix of F , from which we can obtain fast algorithms. In fact, in our applications of quantum communication protocols, we also consider k-party protocols, and our algorithms rely on the approximate low-rank decomposition of the tensor of the corresponding communication problem. To the best of our knowledge, this is the first time that approximate tensor rank is used in algorithm design (approximate rank has been used before, see e.g. [11, 18, 13, 12] and the corresponding related works section).2 1 A low probabilistic rank implies a probabilistic low-rank decomposition of the matrix. 2 We remark that a concurrent work [52] makes algorithmic use of non-negative tensor approximate rank to construct an optimal data structure for the succinct rank problem. L. Chen and R. Wang 23:3 1.1 Quantum Communication Protocols and Deterministic Approximate Counting Our first result is a generic connection between quantum communication protocols and deterministic approximate counting algorithms. I Theorem 1. (Informal) Let X , Y be finite sets and f : X × Y → {0, 1} be a Boolean function. Suppose f has a quantum communication protocol P3 with complexity C(P) and error ε. Then there is a classical deterministic algorithm C that receives A ⊆ X ,B ⊆ Y as input, and outputs a number E such that X f(x, y) − E ≤ ε · |A| · |B|. (x,y)∈A×B Furthermore, C runs in (|A| + |B|) · 2O(C(P)) time. We remark here that there is a simple randomized algorithm running in sub-linear time via random-sampling. Thus the above algorithm is indeed a derandomization of that randomized algorithm. The above theorem can also be easily generalized to the (number-in-hand) k-party case. See Section 2.5 for the definition of the multiparty quantum communication model. I Theorem 2. (Informal) Let X1, X2,..., Xk be finite sets and f : X1, X2,..., Xk → {0, 1} be a Boolean function. Suppose f has a k-party quantum communication protocol P with complexity C(P) and error ε. Then there is a classical deterministic algorithm C that receives X1 ⊆ X1,X2 ⊆ X2,...,Xk ⊆ Xk as input, and outputs a number E such that k X Y f(x1, x2, . , xk) − E ≤ ε · |Xi|. x1∈X1,x2∈X2,...,xk∈Xk i=1 O(C(P)) Furthermore, C runs in (|X1| + |X2| + ... + |Xk|) · 2 time. Sketching Algorithms In fact, Theorem 2 implies a stronger sketching algorithm. Given subsets X1,X2,...,Xk, O(C(P)) the algorithm first computes a w = 2 size sketch ski from each Xi in O(|Xi| · w) time deterministically, and the number E can be computed from these ski’s in O(k · w) time. The sketch computed by the algorithm is in fact a vector in Rw, and it satisfies a nice additive property. That is, the sketch of X1 t X2 (union as a multi-set) is simply sk(X1) + sk(X2). Applying existing quantum communication protocols, we obtain several applications of Theorem 1 and Theorem 2. 1.1.1 Set-Disjointness and Approximate #OV and #k-OV We first consider the famous Set-Disjointness problem (Alice and Bob get two vectors u d and v in {0, 1} correspondingly, and want to determine whether hu, vi = 0), which√ has an efficient quantum communication protocol [1] with communication complexity O( d). 3 We need some technical condition on P, see Corollary 29 for details. I T C S 2 0 1 9 23:4 Classical Algorithms from Quantum and Arthur-Merlin Communication Protocols The corresponding count problem for Set-Disjointness is the counting version of the Orthogonal Vectors problem (OV), denoted as #OVn,d. In this problem, we are given two sets of n vectors S, T ⊆ {0, 1}d, and the goal is to count the number of pairs u ∈ S, v ∈ T such that hu, vi = 0.